Question 4 and 5, Exercise 6.2

Solutions of Question 4 and 5 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

How many $3$-digit even numbers can be formed from the digits $1,2,3,4,5,6,$ if digits are not repeated?

Solution.

We have to sue $3$ digits out of given $6$ digits to make a F3-digit number.
To ensure the created number is even we have to choose the right most digit of number to be even.
Case $\mathrm{I}:$ If unit digit (right most digit) of number is $2$.
$$\underline{ },\underline{ },\underline{2}$$ Possible arrangement of remaining two digits $={ }^{5} P_{2}=20$
Case II: If unit digit is $4$ i.e., $$\underline{ },\underline{ },\underline{4}$$
Possible arrangement of remaining two digits $={ }^{5} P_{2}=20$
Case III: If unit digit is $6$ i.e., $$\underline{ },\underline{ },\underline{6}$$
Possible arrangement of remaining two digits $={ }^{5} P_{2}=20$
$$\text{Total possible 3-digit even numbers }=20+20+20=60$$

How many $7$-digits mobile number can be made using the digits $0$ to $9$, If each number start with $5$ and no digit is repeated?

Solution.

After fixing $5$ as first digits, we have to arrange $6$ digits out of $9$ remaining given digits.
Possible arrangements $={ }^{9} P_{6}=60480$

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